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1. CMB Online first
Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors In a previous paper the authors developed an intersection theory for
subspaces of rational functions on an algebraic variety $X$
over $\mathbf{k} = \mathbb{C}$. In this short note, we first extend this intersection
theory to an arbitrary algebraically closed ground field $\mathbf{k}$.
Secondly we give an isomorphism between the group of Cartier
$b$-divisors on the birational class of $X$
and the Grothendieck group
of the semigroup of subspaces of rational functions on $X$. The
constructed isomorphism moreover
preserves the intersection numbers. This provides an alternative point
of view on Cartier $b$-divisors and their intersection theory.
Keywords:intersection number, Cartier divisor, Cartier b-divisor, Grothendieck group Categories:14C20, 14Exx |
2. CMB 2013 (vol 57 pp. 188)
A Characterization of Bipartite Zero-divisor Graphs In this paper we obtain a characterization for all bipartite
zero-divisor graphs of commutative rings $R$ with $1$, such that
$R$ is finite or $|Nil(R)|\neq2$.
Keywords:zero-divisor graph, bipartite graph Categories:13AXX, 05C25 |
3. CMB 2012 (vol 57 pp. 326)
On Zero-divisors in Group Rings of Groups with Torsion Nontrivial pairs of zero-divisors in group rings are
introduced and discussed. A problem on the existence of nontrivial
pairs of zero-divisors in group rings of free Burnside groups of odd
exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of
zero-divisors are also found in group rings of free products of groups
with torsion.
Keywords:Burnside groups, free products of groups, group rings, zero-divisors Categories:20C07, 20E06, 20F05, , 20F50 |
4. CMB 2011 (vol 56 pp. 407)
On Domination in Zero-Divisor Graphs We first determine the domination number for the zero-divisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zero-divisor graph of any
commutative artinian ring. Finally, we extend some of the results
to non-commutative rings in which an element is a left
zero-divisor if and only if it is a right zero-divisor.
Keywords:zero-divisor graph, domination Categories:13AXX, 05C69 |
5. CMB 2008 (vol 51 pp. 3)
6. CMB 2000 (vol 43 pp. 239)
On the Number of Divisors of the Quadratic Form $m^2+n^2$ For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 |